This section is intended to introduce various aspects of the art, which may be associated with exemplary embodiments of the present invention. This discussion is believed to assist in providing a framework to facilitate a better understanding of particular aspects of the present invention. Accordingly, it should be understood that this section should be read in this light, and not necessarily as admissions of prior art.
Walther's law, named after the geologist Johannes Walther, states that the vertical succession of facies reflects lateral changes in environment. When a depositional environment migrates laterally, sediments of one depositional environment come to lie on top of another. This is the basic stratigraphic concept, integrated with seismic information, allowing petroleum geologists to correlate and extrapolate the detailed geological information obtained from vertical wells throughout the large volume of three-dimensional (3-D) subsurface reservoirs.
Well logs are the most commonly available data in all the wells and are obtained with the use of known well logging tools. When a sufficient number of wells are available, coupling with core data, the interpretation of well-log patterns using sequence stratigraphy concepts provides ultra-high-resolution chronostratigraphic framework for subsurface correlation (J. C. Van Wagoner, et al., 1990, Siliciclastic Sequence Stratigraphy in Well Logs, Cores, and Outcrops: Concepts for High-Resolution Correlation of Time and Facies in Methods in Exploration Series of AAPG book No. 7).
Well-log signal styles, trends and breaks are used to delineate reservoir intervals. These intervals correspond to genetically related geological packages which are used to sketch the framework of the 3-D geobodies. The constituents of the geobodies characterized by internal stacking patterns within intervals are described by well-log curve shapes to reflect the vertical evolution of depositional facies and rock properties.
Well-log correlations include the following three steps:                (1) Subdivide well logs into genetic intervals (or packages) to identify the hierarchy and significance of the stratigraphic boundaries (or framework surfaces);        (2) Characterize stacking patterns between surfaces (or within intervals) to reflect the vertical evolution of depositional facies and rock properties; and        (3) Correlate the well logs to build chronostratigraphic framework based on sequence stratigraphic concept (system tracts, etc.) and information from all sources of data such as seismic, logs, cuttings, cores, etc.        
The interpretation in the last step of well-log correlation is complicated, geological environment dependent and often subjective. It requires expert knowledge, geological concept, and integration of various data including seismic, logs, cuttings, cores, etc. (P. Vail & W. Wornardt Jr., 1991, An Integrated Approach to Exploration and Development in the 90's: Well Log-Seismic Sequence Stratigraphy Analysis, in Transactions Gulf Coast Association of Geological Societies, Vol. XLI). This process can only be automated if one is able to quantify all sources of information as proficient as human's interpretation including expert knowledge and geological concept, which currently is still an interpretive observation, based natural science. Most of the automated well-log correlation techniques were based on single-curve-shape similarity and simple expert rules (U.S. Pat. No. 7,873,476B2, U.S. Pat. No. 7,280,932B2, U.S. Pat. No. 6,012,017A). They can only apply to nearby wells penetrating geobodies of minor geological variations in a specific geological environment. Interpreter's complex and implicit qualitative information from all sources of data is difficult to integrate and translate into algorithms. Up to now, well-log correlation process is still performed manually by stratigraphers.
Quantification of well-log patterns could provide fast, objective and consistent interpretation. Attempts to automate this process have been published since 1960's. Some authors focused on developing algorithms zoning (or segmenting) the well logs, others directly focused on correlating the wells based on similarity of well-log shapes and patterns. Several articles provided good overviews of well-log zonation and well-log correlation techniques: B. R. Shaw & J. M. Cubitt, 1979, Stratigraphic Correlation of Well Logs: An Automated Approach, in Computer & Geology, Vol. 3; I. B. Hoyle, 1986, Computer Techniques for the Zoning and Correlation of Well-Logs, in Geophysical Prospecting 34, 648-664; J. H. Doveton, 1994, Chapter 6: Lateral Correlation and Interpretation of Logs, in Geologic Log Analysis Using Computer Methods, AAPG book; S. M. Luthi, 2001, Section 3.4 Well Correlation, in Geological Well Logs, Springer-Verlag.
Well-Log Zonation
Statistical Approach
Well-log zonation is a critical pre-processing step in well-log correlation. Statistical segmentation (or zonation) techniques received considerable attention in the 1900's literature. However, this approach simply subdivides the well logs into intervals without the capability of identifying the hierarchy and significance of stratigraphic boundaries from log signals. Its fundamental principle is to minimize the variance within each zone and maximize the variance between the zones. Many variations of statistical approach have been published and improved throughout several decades, and they differ from each other in statistical criteria, uni- or multi-variate models, and optimization techniques, etc.: J. D. Testerman, 1962, A Statistical Reservoir-Zonation Technique, in SPE 286; D. Gill, 1970, Application of a Statistical Zonation Method to Reservoir Evaluation and Digitized-Log Analysis, in AAPG Bulletin, Vol. 54. No. 5; D. M. Hawkins & J. A. ten Krooden, 1979, A Review of Several Methods of Segmentation, in Computer & Geology, Vol 3; C. H. Mehta et al., 1990, Segmentation of Well Logs by Maximum-Likelihood Estimation, in Mathematical Geology, Vol. 22, No. 7; D. R. Velis, 2007, Statistical Segmentation of Geophysical Log Data, in Math Geol (2007) 39: 409-417.
Multiscale Wavelet Transform Approach
In the last two decades, methods based on multiscale wavelet transform techniques for identifying multiscale singularities (or discontinuities/edges) in signals became popular in the literature. The mathematical formalism of the continuous wavelet transform was first introduced by Morlet in 1981 and 1982 (J. Morlet, 1981, Sampling theory and wave Propagation, Proc. 51st Ann. Intern. Meeting of Soc. of Exp. Geophysicists; Morlet et al., 1982, Wave propagation and sampling theory-Part I and Part II, in Geophysics Vol 47, no. 2 pp. 203-221 & 222-236).
Witkin in 1984 presented a “scale-space filtering” technique using derivatives of Gaussian (DOG) multiscale transform to generate scale-space image from a one-dimensional signal (A. P. Witkin, 1984, Scale-Space Filtering, IEEE ICASSP), then described a “coarse-to-fine tracking” method to locate singular points (i.e. zero-crossings in the second derivative) at coarse scales and track them down to fine scales on a scale-space image, and finally represented the hierarchy of signal boundaries by an “interval tree” to subdivide a signal with an optimal scale at each section. Vermeer and Alkemade applied Witkin's method to segment well logs in different scales (P. L. Vermeer & J. A. H. Alkemade, 1992, Multiscale Segmentation of Well Logs, Mathematical Geology, 24-1). Mallat and Hwang in 1992 formulated this concept into a popular method: Wavelet Transform Modulus Maxima (WTMM) to characterize the singular behavior of functions (S. Mallat and W. L. Hwang, 1992, Singularity detection and processing with wavelets, in IEEE Trans. Inf. Theory, vol. 38, no. 2, pp. 617-643). DOG wavelets are real-valued functions. Tu and Hwang in 2005 extended WTMM method to complex-valued wavelets (C. L. Tu & W. L. Hwang, 2005, Analysis of Singularities from Modulus Maxima of Complex Wavelets, in IEEE Trans. Inf. Theory, vol. 51, no. 3, pp. 1049-1062).
Robail et al. in 2001 proposed a method using the scale-depth phase image of a complex-valued Morlet continuous wavelet transform (CWT) to process well-log signals (F. Robail et al., 2001, Sedimentary Bodies Identification Using the Phase Coefficients of the Wavelet Transform, in SPWLA 42nd Ann. Log. Symp. paper VV). As the well log signals are measured in depth, the wavelet transformed image is a scale-depth representation of well logs. The features exhibited on a CWT phase image of a well-log signal bring out multiscale intervals (or geological packages) and their hierarchical relationships. However it is not a trivial task to automatically extract the features revealed by the CWT phase image. Similar to Witkin's process, Robail et al. tracked the phase lines (where the phase values change signs) on the CWT phase image, and then represented the boundary hierarchy with Witkin's interval tree to subdivide the well log signals. More details about their method is discussed below.
U.S. Pat. No. 6,366,859 (“Method of Detecting Breaks in Logging Signals Relating to a Region of a Medium”) describes a simple method for detecting breaks in well-log signals. The technique first calculates the absolute value of the mean gradient of the characteristic quantity of the wavelet transform of well logs, and then selects the peaks as breaks.
Hruska et al. in 2009 used discrete wavelet transform (DWT) with Daubechies' wavelet to segment the borehole image logs (M. Hruska et al., 2009, Automated Segmentation of Resistivity Image Logs using Wavelet Transform, in Math. Geosci. 41: 703-716). CWT can accurately locate the boundaries in a signal, however, the boundaries generated from DWT can only occur at positions which are multiples of 2L, where L is the scale used, i.e. the DWT boundaries do not always lie in the correct location.
Characterization of Curve Patterns Between Surfaces
The curve-shape patterns of well logs have been classified and related to sedimentological phenomena for sand-shale formations (J. C. Van Wagoner, et al., 1990). By analyzing the lower and upper bedding contacts and the shape of the curve, the log curve shapes can be characterized in shapes of cylinder, bell, funnel, egg, smooth, serrated, concave, convex, etc. (O. Serra & L. Sulpice, 1975, Sedimentological Analysis of Shale-Sand Series from Well Logs, in SPWLA 16th Annual Logging Symposium, paper W; O. Serra & H. T. Abbott, 1982, The Contribution of Logging Data to Sedimentology and Stratigraphy, in SPE Journal).
The quantification of curve shapes has not received much attention in the literature. Simple linear regression of well logs was commonly used for determining fining-upward and coarsening-upward sequences. Statistical methods have been proposed in the literature (C. Reiser, 1998, Identification des Corps Sedimentaires et des Sequences Stratigraphiques par l'Analyse Numerique des Formes Diagraphiques, French PhD Thesis, University of Lyon I; F. Robail et al., 2001, SPWLA paper VV). They first extracted statistical parameters such as the sign and absolute value of the slope and kurtosis and skewness of the curves defined by bedding intervals, then Reiser represented the parameters in spider plots for each type of curves, and Robail et al. partitioned the parameters into groups. These statistical methods provide “relative” quantification of curve shapes, and their representations (spider plots, colored patterns, etc.) are impractical for multi-well interpretation and/or to extend the knowledge to different fields.